Abstract
Experiments on the fractional quantized Hall effect in the zeroth Landau level of graphene have revealed some striking differences between filling factors in the ranges $0<|\ensuremath{\nu}|<1$ and $1<|\ensuremath{\nu}|<2$. We argue that these differences can be largely understood as a consequence of the effects of terms in the Hamiltonian which break SU(2) valley symmetry, which we find to be important for $|\ensuremath{\nu}|<1$ but negligible for $|\ensuremath{\nu}|>1$. The effective absence of valley anisotropy for $|\ensuremath{\nu}|>1$ means that states with an odd numerator, such as $|\ensuremath{\nu}|=5/3$ or 7/5, can accommodate charged excitations in the form of large-radius valley skyrmions, which should have a low energy cost and may be easily induced by coupling to impurities. The absence of observed quantum Hall states at these fractions is likely due to the effects of valley skyrmions. For $|\ensuremath{\nu}|<1$, the anisotropy terms favor phases in which electrons occupy states with opposite spins, similar to the antiferromagnetic state previously hypothesized to be the ground state at $\ensuremath{\nu}=0$. The anisotropy and Zeeman energies suppress large-area skyrmions, so that quantized Hall states can be observable at a set of fractions similar to those in GaAs two-dimensional electron systems. In a perpendicular magnetic field $B$, the competition between the Coulomb energy, which varies as ${B}^{1/2}$, and the Zeeman energy, which varies as $B$, can explain the observation of apparent phase transitions as a function of $B$ for fixed $\ensuremath{\nu}$, as transitions between states with different degrees of spin polarization. In addition to an analysis of various fractional states from this point of view and an examination of the effects of disorder on valley skyrmions, we present new experimental data for the energy gaps at integer fillings $\ensuremath{\nu}=0$ and $\ensuremath{\nu}=\ensuremath{-}1$, as a function of magnetic field, and we examine the possibility that valley skyrmions may account for the smaller energy gaps observed at $\ensuremath{\nu}=\ensuremath{-}1$.
Highlights
This energy will scale the same way as the Zeeman energy, proportional to B, as previously remarked. (We note that the relative importance of these terms can be altered, by application of a parallel field, which will increase the Zeeman energy but not affect the valley anisotropy energies.) Because the energies due to Coulomb interactions scale as B1/2, we see that the ratio between these and the valley anisotropy and Zeeman terms will change when the magnetic field is varied at fixed filling factor, and transitions between different ground state configurations could occur as a result
Taking into account the implications of Eq(7), we find that if the lowest energy state at f = 2 is the AF state, the electron skyrmion at f = 1 with lowest anisotropy energy will be one where the hyperspin state far from the skyrmion has electrons in one valley and spin aligned with the magnetic field, while at the center of the skyrmion the electrons sit in the opposite valley and have reversed spin
That the optimum ground states for 0 < f < 1 were obtained when the two constituent hyperspin states had opposite valley indices but both spins aligned with the magnetic field
Summary
To understand the underlying symmetries, let us start with the microscopic Hamiltonian of the zeroth LL. We note that LL mixing will lead to some renormalization of the SU(4) invariant Coulomb interaction, on the scale of the magnetic length, but we ignore such effects here. The valley anisotropy terms in our model, (4) and (5), have effect only when two electrons coincide in space This reflects the fact that on the microscopic scale, sensivity to the electron valley or sublattice occurs only when two electron are separated by a distance of the order of the graphene lattice constant, which is much smaller than the magnetic length. We remark that in the zeroth Landau level, the electrons in opposite valleys are confined to opposite sublattices With these assumptions the contributions of gz and g⊥ to the energy per flux quantum, for a fixed value of ν, will scale as gz/(2πlB2 ) and g⊥/(2πlB2 ), respectively. This energy will scale the same way as the Zeeman energy, proportional to B, as previously remarked. (We note that the relative importance of these terms can be altered, by application of a parallel field, which will increase the Zeeman energy but not affect the valley anisotropy energies.) Because the energies due to Coulomb interactions scale as B1/2, we see that the ratio between these and the valley anisotropy and Zeeman terms will change when the magnetic field is varied at fixed filling factor, and transitions between different ground state configurations could occur as a result
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
